Some Notions of Separability of Metric Spaces in $\mathbf {ZF}$ and Their Relation to Compactness
Volume 64 / 2016
Bulletin Polish Acad. Sci. Math. 64 (2016), 109-136
MSC: 03E25, 54D65, 54D30.
DOI: 10.4064/ba8087-12-2016
Published online: 20 December 2016
Abstract
In the realm of metric spaces we show in $\mathbf {ZF}$ that:
(1) Quasi separability (a metric space $\mathbf {X}=(X,d)$ is quasi separable iff $\mathbf {X}$ has a dense subset which is expressible as a countable union of finite sets) is the weakest property under which a limit point compact metric space is compact.
(2) $\omega $-quasi separability (a metric space $\mathbf {X}=(X,d)$ is $\omega $-quasi separable iff $\mathbf {X}$ has a dense subset which is expressible as a countable union of countable sets) is a property under which a countably compact metric space is compact.
(3) The statement “Every totally bounded metric space is separable” does not imply the countable choice axiom $\mathbf {CAC}$.
